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Theorem csbiebt 3915
Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3919.) (Contributed by NM, 11-Nov-2005.)
Assertion
Ref Expression
csbiebt ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbiebt
StepHypRef Expression
1 elex 3517 . 2 (𝐴𝑉𝐴 ∈ V)
2 spsbc 3788 . . . . 5 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴𝐵 = 𝐶)))
32adantr 481 . . . 4 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴𝐵 = 𝐶)))
4 simpl 483 . . . . 5 ((𝐴 ∈ V ∧ 𝑥𝐶) → 𝐴 ∈ V)
5 biimt 362 . . . . . . 7 (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ (𝑥 = 𝐴𝐵 = 𝐶)))
6 csbeq1a 3900 . . . . . . . 8 (𝑥 = 𝐴𝐵 = 𝐴 / 𝑥𝐵)
76eqeq1d 2827 . . . . . . 7 (𝑥 = 𝐴 → (𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))
85, 7bitr3d 282 . . . . . 6 (𝑥 = 𝐴 → ((𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
98adantl 482 . . . . 5 (((𝐴 ∈ V ∧ 𝑥𝐶) ∧ 𝑥 = 𝐴) → ((𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
10 nfv 1908 . . . . . 6 𝑥 𝐴 ∈ V
11 nfnfc1 2984 . . . . . 6 𝑥𝑥𝐶
1210, 11nfan 1893 . . . . 5 𝑥(𝐴 ∈ V ∧ 𝑥𝐶)
13 nfcsb1v 3910 . . . . . . 7 𝑥𝐴 / 𝑥𝐵
1413a1i 11 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥𝐶) → 𝑥𝐴 / 𝑥𝐵)
15 simpr 485 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥𝐶) → 𝑥𝐶)
1614, 15nfeqd 2992 . . . . 5 ((𝐴 ∈ V ∧ 𝑥𝐶) → Ⅎ𝑥𝐴 / 𝑥𝐵 = 𝐶)
174, 9, 12, 16sbciedf 3816 . . . 4 ((𝐴 ∈ V ∧ 𝑥𝐶) → ([𝐴 / 𝑥](𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
183, 17sylibd 240 . . 3 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) → 𝐴 / 𝑥𝐵 = 𝐶))
1913a1i 11 . . . . . . . 8 (𝑥𝐶𝑥𝐴 / 𝑥𝐵)
20 id 22 . . . . . . . 8 (𝑥𝐶𝑥𝐶)
2119, 20nfeqd 2992 . . . . . . 7 (𝑥𝐶 → Ⅎ𝑥𝐴 / 𝑥𝐵 = 𝐶)
2211, 21nfan1 2193 . . . . . 6 𝑥(𝑥𝐶𝐴 / 𝑥𝐵 = 𝐶)
237biimprcd 251 . . . . . . 7 (𝐴 / 𝑥𝐵 = 𝐶 → (𝑥 = 𝐴𝐵 = 𝐶))
2423adantl 482 . . . . . 6 ((𝑥𝐶𝐴 / 𝑥𝐵 = 𝐶) → (𝑥 = 𝐴𝐵 = 𝐶))
2522, 24alrimi 2206 . . . . 5 ((𝑥𝐶𝐴 / 𝑥𝐵 = 𝐶) → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶))
2625ex 413 . . . 4 (𝑥𝐶 → (𝐴 / 𝑥𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶)))
2726adantl 482 . . 3 ((𝐴 ∈ V ∧ 𝑥𝐶) → (𝐴 / 𝑥𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶)))
2818, 27impbid 213 . 2 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
291, 28sylan 580 1 ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1528   = wceq 1530  wcel 2107  wnfc 2965  Vcvv 3499  [wsbc 3775  csb 3886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-v 3501  df-sbc 3776  df-csb 3887
This theorem is referenced by:  csbiedf  3916  csbieb  3917  csbiegf  3919
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